I am a mathematical flirt. The Jesuit who taught me was far too clever to understand that I needed my sixth-form mathematics explained in words of one syllable, and preferably in terms of apples and oranges. But I was fascinated by a discipline which could not only describe the world yet could discover otherwise undetectable truths.
Are mathematics inherent in reality, or just our human grasp of the fundamental order of creation? I do not know, nor, as far as I can tell, does anyone else. But today I just want to highlight examples of the mathematics which I find especially appealing.
I am now about to give you the details of my credit card. It measures 55mm by 85mm. That means that Euclid and Plato would have approved, and so would the designer of the Acropolis, for these proportions are the key to its structures. The ratio of approximately eight to 13 is known as the golden ratio or, as Pacioli in the 15th century called it, the “divine proportion”. And that is a good name because that ratio is manifest in so many areas in art, mathematics and nature, suggesting its relationship to creation. If heaven has gates they will certainly satisfy the divine proportion.
Take a line and bisect at the point where the proportion of the original line to the longer section is the same as the proportion of the longer section to the shorter (got that clear?) and you have the divine proportion.
The human eye finds that proportion aesthetically satisfying. A look at Renaissance and traditional painting will show in how many ways the artist uses it to achieve a pleasing composition which draws our attention to where he directs us.
Take a line of numbers. This one is called the Fibonacci sequence. Here, the following sequence is discovered as the sum of the two preceding numbers. Thus: 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. Only of casual interest? Not quite. Notice 8 and 13 – which is once more the ratio of the divine proportion. Indeed, successive Fibonacci numbers confirm this.
And it turns out to be everywhere. It appears in DNA and the hornplates of a turtle, the anatomy of a spider and the breeding of rabbits. It is the ratio involved in the division of tree branches and in the polyfurcation of veins. It is as if the mind of God, the structure of creation, and the mind of man were all connected by a single principle of beautiful harmony which lies at the heart of all existing things.
GIGO means Garbage In, Garbage Out. Wise words for many endeavours but not necessarily true. If you can remember how, solve this equation: i2 = −1 by taking the square root of both sides. The answer is i = √−1. Only it can’t be resolved because any number multiplied by itself is positive; it cannot be negative. √−1 is an “imaginary” number. But before you throw it away as garbage, just check on your digital camera. You will find that the equation which compresses the pixels contains such imaginary numbers. It has been discovered that many equations, almost too complex to be solved directly, can be simplified quickly by such numbers. Many of the operations in the modern world depend on using imaginary numbers.
Of course the answer would be useless if it contained √−1. But it doesn’t. In lay terms, imaginary numbers act as a scaffold brought on to assist, and then quietly removed. You and I know that the scaffold was there, we just didn’t see it.
What proportion of Catholics, at given ages, go regularly to Sunday Mass? Because I can never investigate them all I will have to use a sample. Leaving aside the (fascinating) issue of choosing the right sample and working out how the information will be obtained, I will still have to measure the accuracy of my answers. It may help me to go for an average, remembering that mean, median and modal averages may each be different. But I will also need to have a measurement of whether the distribution is broad or narrow. The standard deviation, which tells me that, will also indicate how my accuracy relates to sample size.
I may need to check my correlations, too. For example, is there a strong or weak correlation between age and Mass attendance? There is a calculation which tells us that, too.
I know that mathematics cannot give certainty in such matters but I am thrilled at the thought that we can calculate our degree of uncertainty. And when people say to me: “But they only asked 1,000 people. That can’t give a reliable answer.” I merely tap my nose and say: “It all depends.”
Yet ultimately all mathematical bets are off. Kurt Gödel may be one of our few contemporary personalities who will be remembered in 1,000 years. He showed that any formal system of mathematics that includes a modicum of arithmetic is incomplete, and there will always be true statements that cannot be proved within the system. Gödel died in 1978 of “malnutrition and inanition” caused by “personality disturbances”. Which shows that mathematics can drive you mad.
But the journey can be fun. I gave The Math Book and The Physics Book (by Clifford A Pickover and published by Sterling) to two grandsons (sixth form and undergraduate). They have reported with enthusiasm on these beautifully illustrated books – sufficiently interesting to them to be permanently at the bedside, and I pass on their firm recommendation to you.