Maths, we know, is today in a sorry condition, and on all levels, from uninspiring school teaching, to mathematical logic which still can’t solve Russell’s Paradox or the Continuum Hypothesis.  Maths used to be the central motif which was responsible for Western Culture and the Modern World.  Now Western Culture has become a whipping boy, and the Modern World seems to be heading for dystopia —partly to be brought about by so-called super-human AI, and partly resulting from an unsustainably hot planet. How can we expect young people to live normally in such a gloomy mindframe?

Maths used to be the most superior of the superior elites, but now it is an open question whether it still counts as an ‘Elite’ at all.     

Q: When does an Elite cease to be an Elite?

A1: When it stops believing its own logos.

A2:When the general public stops treating it as an Elite.

In the case of the Elite of Higher Mathematics both these possible answers make sombre reading.

On A2 the box has clearly already been ticked.  During the recent heated debate over whether AI is actually intelligent, the media have not even bothered to consult the higher mathematicians. 

Ironically the mathematical element in computing is a major reason why it can look ‘intelligent’ —which is doubly ironic, because the leadership of maths let the computer sector take full credit for all applications of maths as early as the 1960s.

And the general public have been mostly treating mathematics as obsolete —as a low-tech, redundant precursor to the computer— for quite a while.  There is of course a subset of the public consisting of people who still buy books vicariously describing the obscure triumphs of higher maths: in many cases they are relatives or friends of mathematicians, or people who faintly recall being good at maths at school. 

On A1 much self-doubt has already been aired. Henri Poincare declared that the transfinite was a disease from which mathematics would eventually recover. Morris Kline described mathematics as an ‘uncertain science’, and he showed just how screwed-up it had become. Stanley Ulam pointed to his dilemma, which had terminally muddied the previously clear waters of hyper-abstract mathematic truth.  At least Kline credited maths with still being a ‘science’, whereas many of the gurus of the subject have been saying since 1900 that higher mathematics is really an ‘intellectual artform’, ‘a supreme intellectual culture immensely valuable for its own sake’.  

But this kind of rhetorical bravado sounds pretty empty now that the general public have lost their support. If they think about it at all, they tend to regard it as an in-house culture with a ludicrously exaggerated, self-serving concept of its own importance.

What can be done about this parlous situation?

There is a way out of today’s crisis. It was previously variously ignored, rubbished, brushed-aside and ridiculed.  It involves recognising that mathematics is a humanly constructed tool which can act as the ‘Pathfinder for Progress’.

The notion which was totally dominant for twenty-five centuries was that maths was a kind of secular religion. (Like religion it dealt in ‘Right’ and ‘Wrong’. Like religion it claimed to explore eternity.)  Its high priests were quite arrogant. At the time of Pythagoras they taught that God had made a mistake when he created the universe!

Could anything be more ridiculous?

We need to switch to a more sensible point of view   —we should have made this switch in the 1960s. A new age has dawned:

1. The feeling that maths has meaning —as appreciated by the average person— has always rested on its wonderful instrumental uses. In the early days they were mainly about military logistics and achieving geometric elegance in buildings.

2. These instrumental uses involve a lot of reasoning. They add up to a way of illuminating the predictable future.

3. As G H Hardy said in his Mathematician’s Apology, the most useful part of maths was, for many centuries, the pure maths which occasionally enabled instrumental analyses to deliver practical answers. Things like algorithms to solve equations, trigonometry, difference methods, logarithms… were the most useful bits of pure mathematics which delivered these outcomes.

4. The computer arrived around 1960. It automated mathematic processes and made the subject a thousand times more powerful than it had previously been.

5. Since the arrival of the digital computer and the automation of its processes, maths has been able to deliver answers across a vast arc of practical applications. 

6. We urgently —very urgently— need a revolution in teaching maths, one which treats it as a wonderful source of practical illumination. It should be approached via curiosity about new projects, innovations, schemes and inventions, not as an austere quasi-religious monklike escape from tiresome banal reality. Maths, in a word, needs to be seriously presented as being about the real world, not about an imagined painless castle in the sky.   

CHRISTOPHER ORMELL 1^st August 2023

If you would like an online copy of the P E R Narrative Maths Manifesto, send an email to per4group@gmail.com asking for this.