In order to understand maths you have to see it as a language and then to study/learn other languages and contrast them with maths. The more of these other languages you learn the deeper will be your understanding of maths, so, once you have studied music, visual communication (art), English, Japanese, computer languages and bird song then you will begin to understand maths. (see my post: Understanding).

But, to start with, it is useful to class languages according to the semantics/syntax complementarity relationship, which says that the more syntax a language has, then the less semantics (meaning), and vice-versa. Maths then lies at the extreme of having loads and loads of rules, i.e. syntax, and being, therefore, virtually devoid of meaning. This instantly places maths as a language suitable for machines, but useless for any communication between human beings.

Bertrand Russell did a very nice ‘housework’ job on mathematics in his book, Principia Mathematica i.e. he tidied everything up and put it in its proper place and cleaned out the rubbish – something which happens all to rarely in the academic world – not surprisingly since housework is so despised.

Anyway, the essence of Russell’s work is this: that you create mathematics (the Big Bang of mathematics is) by postulating the existence of zero and infinity, define what ‘numbers’ are (set theory), and define the rules that govern the interactions of numbers i.e addition is commutative ( a+b=b+a) etc.

That gets you arithmetic, and then, to go on to the other types of maths, such as algebra, trigonometry, and geometry and so on, you just add another rule or two, and another variable. For example, to get from algebra to vector algebra you add another variable, or dimension, that of ‘direction’ along with the new rules required to define how to operate it. In other words, maths forms a sort of evolutionary tree that starts with arithmetic and then grows and keeps on branching, so that every form of mathematics is a branch of that tree, and the tree is still growing, and if you want to add a branch you have to add a new variable or two, and a new rule or two, to an existing branch and bobs your uncle.

It’s rather like the relationship between games like draughts, backgammon and chess: you can start with any one of these games and ‘evolve’ it into any of the others by a process of adding or subtracting pieces and rules that govern their movements, and allowing the board to evolve too.

In fact, mathematics in its practice is more like a game that a high level language like English.

I read a book recently which featured a chess grand master. The reader was obviously expected to be in awe of such an exalted being, someone of such amazing intelligence. Almost with reverence the author told of how he held in his memory every game that had ever been played that he had seen or read about etc. Well, that’s about it, really. As with chess, in maths you can get by on memory i.e. you do not even need to master the rules, let alone understand anything (understanding chess is understanding warfare), and it is obvious when you hear them talking, or read their work, that many top mathematicians, let alone understand anything, have not even mastered the rules but are operating on memory alone.

Physics and the other sciences relate to one another in this evolutionary tree mode also, with physics as the simplest, the ‘arithmetic’ of science. It is in physics that you find the Big Bang of the sciences: i.e. postulate the existence of ‘space’ and ‘time’ and define your variables, in this case matter and energy, and then define the rules that govern their interactions, that is, define the most basic laws of physics. The ‘higher level’ sciences, like chemistry and biology etc are reached by adding variables and rules.

So the top scientists, like the top mathematicians and chess players can operate purely on memory, not even having mastered the skills of the game let alone understanding anything, and their writings give the game away – if you have the mind to see it i.e. if, like them, you are getting by on memory alone, then you are not going to have the ability to see it.

This then turns the world upside down, yet again: that is too say that the ‘top intellects’ are not to be found among physicists or mathematicians – and all I have said above goes for philosophy too. So where are the ‘top intellects’ to be found? Well, it’s rather like Lord of the Rings, where the Ring Bearer could not be one of the ‘big’, important people, but had to be a hobbit, one of the ‘little’ people that no-one hardly ever notices.

The alarming thing is that it is these people of small intellect who are largely in positions of control in our world. Why and how? In the words if W.B. Yeats:

“The worst are full of passionate intensity

While the best lack all conviction.”

So we have these dummies designing and running nuclear power stations, chemical plants, arms and armaments – including the nuclear variety – and taking over control of our lives with their climate change, and environmentalism, and conservation and their solutions to the problems of famine and poverty and disease and the like – not to mention trying to control ‘the economy’.

Not that I am trying to say that the world is not in a bad way. It is, but not for the reasons they think, and they are, in fact, more the problem than the solution (read my post: Apocalypse Now!).