THE INSTITUTE OF MATHEMATICS
AND ITS APPLICATIONS

Editorial

What do you understand by a champion? There is a motor racing champion, a boxing champion and of course the classical champion – a knight who fought on behalf of a lady, then (to people of my generation) there was champion the wonder horse and of course there is that well known manufacturer of sparking plugs.

Which (if any) of these inspired the government’s response to the report from Sir Peter Williams on the state of mathematics teaching in primary schools is not known. However we are told that every school will have a ‘maths champion’, to be a mentor and coach to their colleagues, as well as being an outstanding teacher for their pupils.

Fear and distaste for mathematics cannot easily be disguised. If teachers themselves achieved only a GCSE grade C at mathematics, it is unlikely to be a subject with which they feel comfortable. At best they will communicate that it is difficult, at worst that it is a subject where little can be achieved.

When it comes to mathematics teaching, Britain is a failing nation. Perhaps task groups of teachers from abroad should be sent in to replace those who have little affinity with the subject.

It was depressing to see political correctness paraded on television in an interview with a teacher who it seemed had been brainwashed into reciting the mantra that if a child cannot do mathematics, the fault lies not with the child but with the teacher; the teacher has not communicated correctly with the child.

All people are born equal but that does not mean that they are all equally capable. Some children could be given the world’s greatest teacher and would still be poor at mathematics. What we need to ensure is that those capable of doing mathematics are not put off. There are many things that do this and in particular the attitude of those adults that surround them.

I suspect we have all been praised at some time or another for our mathematical ability and then treated in a different way from other people. Some children cope well with this but others would do anything rather than be distanced from the company of their peers. We hear of this pressure to a lesser degree in failing schools, where children have been bullied into not achieving so that others, less able but more powerful, are not humiliated.

I don’t know if the Mathematics Champions will be able to deliver all that is expected of them but unless all the teachers in the school (and the parents too) change their attitude to mathematics it will be difficult.

In recent months we have heard the father of the London Mayor advance the idea that because Boris Johnson is exceptionally able in both Latin and Greek he is therefore more than capable of tackling whatever problems running the capital may present.

Mathematicians often do exceptionally well in other areas but mathematics is seldom seen as the reason for their ability. Indeed it takes some courage, in a social gathering among strangers, when asked what you do for a living, to say you are a mathematician. Almost invariably someone in the group will volunteer (with perhaps false regret) the information that they never could understand mathematics when they were at school.

There is also an unspoken subtext. ‘This person must be some kind of a freak. Look at me, I’ve done well in life and never had to bother about mathematics.’ It is a national scandal that it is a badge of ‘coolness’ not to be able to do mathematics. Anti-mathematics is a peculiarly British disease but now the chickens have come home to roost.

Pressure on the curriculum has resulted in mathematics being squeezed between time taken for computing and time taken for statistics. Is it therefore surprising that GCSE mathematics questions are now a pale reflection of what they were 40 years ago?

Logic is the bedrock of mathematics which unfortunately has all but been eliminated from the curriculum by applications of routine formulae and simple techniques of little merit. Mathematics has little to do (and the less, the better) with the construction of bar charts and histograms and similar mind-numbing activity.
 
Why is it that only in mathematics is absence of skill seen to be a positive social advantage? Mathematicians understandably camouflage their activities (the skill that dare not speak its name) whenever they can – analytical engineer, code breaker, security analyst, accountant, financial adviser, actuary, scientific analyst. However none of this will work, already actuaries and accountants are being rumbled. Humour is the weapon – it is said that those who become actuaries are those who found accountancy too exciting! We too can laugh.

People tell me they can spot the mathematician on University Challenge before he or she has even opened their mouths, so it must be something in the demeanour that alerts them, I doubt it is their evident intelligence.

How depressing that young people, not just those who wrestled with logarithmic tables or calculated how long it would take to fill a bath without a plug, but those who learnt about symmetry, number systems and all the stuff that was meant to make mathematics exciting and stimulating, eschew mathematics as a pariah subject and treat those who practice it as people in serious need of getting a life.

Do you believe in the Axiom of Completeness? Perhaps I should put this another way, do you accept the Axiom of Completeness? As you know, you don’t have to; you can reject it and then confine your attention to non-standard analysis. However, to me, non-standard analysis lacks the sublime quality of classical analysis (possibly an acquired taste), where there is subtlety of logic and beauty of proof.

In recent months two large tectonic plates have been struggling to assert their authority and the rumbles have been felt almost everywhere. The two tectonic plates are those which most influence people today - science and religion. Unfortunately, these two giants do not always rub along well with one another.

Indeed a little while back, Mathematics Today carried a flurry of letters concerning the tussle between those who wish to include creationism (in some form or other) in the science curriculum and those who wish to confine things to Darwin’s theory of evolution.

Darwin certainly explains many things but surely not everything, the vast varieties of life forms that exist in the world are certainly more than necessary to exploit the available resources. On the other hand, the phenomenon (unfortunately common throughout history) of a victorious army slaughtering the defeated male population and impregnating the women is consistent with Darwinism for ensuring the survival of the fittest. Indeed this is replicated in the wild where dominant lions kill not only the males but also any cubs they may have sired.

Science and religion have striking similarities; each builds on axioms or assumptions and each uses reason to enunciate broad principles. Moreover, each has within its community members who cannot tolerate the premises of the other community.
Possibly the main difficulty is communication and the use of language. There are two modes of description – scientific and religious, sometimes they are in harmony and sometimes they are not. In modern parlance, science is a bottom up approach while religion is top down. Science is strong on verification but not on ethics whereas religion is strong on ethics but not on verification; science is built on observation, religion is built on belief.

Recently the radio has been airing discussions on this topic and there I heard about the Brights. These are a group of scientists and philosophers who assert that those with religious views are intellectually inferior; to them religion is an anachronism or a prop for those who are incapable of facing reality. On another program I heard it suggested that scientists should be given complete freedom without any ethical constraints being imposed upon them.

What has been somewhat depressing about these spats is the level at which it has been conducted. All too often, it has provided a demonstration of complete misunderstanding between the two sides of what the other is about.

Parliament has agreed to allow scientists to create hybrid embryos. The embryo from a cow will have its contents replaced with human stem cells, these hybrid embryos will be allowed to develop for a short time and the benefits to humanity could be significant. Hope is in prospect that many currently incurable diseases such as Motor Neurone disease and Parkinson’s disease will yield to scientific analysis. How can any rational person object to this?

Well, enquiry is central to science, ‘what if this or what if that?’ So once this technique has been developed (and there can be little doubt that it will be) as sure as eggs are eggs, someone will ask, ‘what would happen if that embryo were allowed to develop to maturity?’ It would be illegal in Britain to allow this but once knowledge is gained (as we have seen with nuclear weapons) it cannot be surrendered – the genie is out of the bottle.

In a few years time the possibility exists that a creature will be born that is 100 - p% human and p% something else, where p > 0. What would be the status of that creature? Would it be murder to kill it or would it only have animal rights? There are sinister possibilities; we may indeed be on the threshold of a Brave New World.

The notion that life began in a primeval swamp had become generally accepted with the belief that all the ingredients for life developed on earth. It was therefore exciting to read the interview with Professor Chandra Wickramasinghe in the April edition of Mathematics Today and the increasing evidence that comets were the agents that brought life to earth.

Comets, it seems, are likely to contain fully developed genetic structures and we have the real possibility that comets stream through the universe occasionally colliding and impregnating planets, if the conditions are right then life can develop. So, did life arrive on earth in some sense flat-packed? Do all comets consist of the same basic materials? Is it possible to trace any of these genetic structures that may have landed on the moon?

The article got me thinking too about what, if anything (scientifically speaking) could be said to be independent of time and space. Clearly Newtonian mechanics and indeed all applied mathematics and engineering depend on the physical universe, as do the sciences. Music depends on sound, art depends on light, so very little can be said to be independent of time and space.

What about number theory? If anything exists, we have the notion of ‘one’ and therefore the natural numbers; prime and composite. So, for instance, Euclid’s celebrated theorem, that there are an infinite number of prime numbers, is something that is independent.

For convenience, I use the word transcendental for mathematics that is independent of time and space. Some abstract mathematics and general topology may be transcendental but not all pure mathematics falls into this category, for anything requiring visualisation is automatically dependent on time and space. It would be interesting to see how pure mathematics partitions into that which is transcendental and that which is not. There are some grey areas of course – what about abstract mathematics that has its genesis in geometry but can be presented in a way that is independent of time and space? To develop these ideas further would require more analysis which … ‘the margin of this page is too narrow to contain.’

Guest Editorial by Charles Evans CMath FIMA
Honorary Secretary IMA (Designate)
Formerly University of Portsmouth