A Bouquet for Gauss by Suman Vaze

I was recently put in front of a class who had been doing some work on Complex Number. They knew about real and imaginary parts, and had done some algebra as well as drawing representations on the complex plane. They had heard of the terms modulus and argument, but had not really done anything with them. I began the lesson by putting Suman Vaze’s painting on the screen.

I left the class for a while to get something i had forgotten, and asked them to study the painting and find the Maths in it. I suggested they concentrate on the first four flowers.

After leaving them for a while, we started to discuss what they had noticed. Various comments came in, and the built on the wisdom of each other. We reached rotation, angles, different ways of arriving at a half turn, adding four equal angles together to do this. We even discussed whether these four solutions were the only ones, and decided that they were if we accepted angles being modulo 2π being equal and only allowed positive rotation.

I then asked them to express this in the language of complex numbers, and we eventually arrived a the fourth roots of -1, which was Gauss PhD thesis. Once they had the feeling of rotation and multiplication, their grasp of modulus and argument was intuitive, and the statement and proof of de Moivre was obvious.

A colleague and I were discussing continuous random variables the other day (yes I know), it occurred to me that the problem students have with this is understanding that the probability of any one value is zero and yet all these zeros add up to one!  This is counter intuitive for students, and is at the heart of the process of integration. In the modern world, well the last 200 years or so, we take a very prudish view of the infinite when we teach mathematics. We are seduced by Cauchy and Wierstrass and their rigorous redefinition of the calculus as being about limits. Of course, mathematically this is absolutely correct, but intuitively it is tough.

It was not always like this. There is evidence that Newton and Leibniz thought in terms of infinitesimals even though they were very suspicious of them. They used them in calculations without defining them, which led to Bishop Berkeley’s famous condemnation of “Ghosts of departed quantities”. Archimedes certainly used the idea of adding up an infinite number of infinitesimals in The Method to get the results he published elsewhere. There is of course Robinson’s wonderful Non Standard Analysis that puts all this in a rigorous setting. New systems of number such as Conway’s show how we can create infinitesimal and infinite numbers if we just wait long enough!

I would suggest that this approach to the calculus is much more intuitive than the usual stress on limits, which can happen at university. It also makes Integration more natural than Differentiation and I would always teach it first. The Fundamental Theorem of Calculus becomes proper result rather than the way integration is defined. See Elvis and his Impersonators discuss the theorem.

Now for the fun part:

Chris Binge’s intuitive demonstration of the unaccountability of the irrationals.

We know that the infinite decimal expansion of a rational number always ends in an infinite recurring series. This may be a series of zeros (as in 1/2) another number (1/3) or a longer sequence (1/7).

Take a 10 sided die, marked with the digits from 0 to 9. Write down “0.” And then start rolling the die, writing down the digits you get. What you are doing is constructing a number between 0 and 1, that is produced entirely by chance. Imagine (this is a thought experiment) the dice being thrown an infinite number of times, or thrown once each by an infinite number of monkeys.

You will produce a random real number between zero and one. Is any one real number any more likely than any other? Obviously not, so they are equally likely.

How likely is it to get ½? O.5000000…   forever? Well, if this isn’t zero then I do not know what else it could be. So the chance of any number is zero.

How likely is any infinitely recurring sequence? Intuitively, a repetition of any sequence is likely to break down eventually – with probability 1. Or at least the probability of any pattern continuing infinitely recedes towards zero. So the probability of a recurring decimal is zero. Hence the probability of a rational number is zero. Almost all reals are irrational!

Desargues by Suman Vaze
If two triangles (in red) are in perspective, then pairs of corresponding sides meet at three points which are collinear. Two triangles are said to be in perspective if lines joining corresponding vertices meet at a point.

…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.

An activity

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.

Creative expression

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.

A tool

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.