These blogs are offered to show the way in which conceptual, reasoning-led maths can generate striking, even sometimes major, results. In an age of awesomely microscopic, awesomely fast, massive digital electronics, authentic, traditional maths cannot compete with the latest hardware —if the contest is simply about the dexterity of manipulative process.  But we are not handle-turners. The heart of the challenge of maths is not to operate technically difficult manipulations.  Maths has made some of its most spectacular forward-leaps   —like Euclid’s demonstration that the set of prime numbers is infinite—  by marshalling the power of rational human insight… or if you prefer, the careful use of transparent reasoning.

The last post (Blog 53) was based on an article in The Spectator (September 2024) page 15 by David Whitehouse. His theme was that there are extraordinarily difficult problems in maths connected with prime numbers. His piece highlighted the still unproven Riemann Hypothesis as being probably the most difficult. He quoted David Hilbert as saying that <<If he woke up after sleeping for 500 years, his first question would be whether the Riemann Hypothesis had been solved>>!

But how important would such a solution be?

Whitehouse makes no reference to the looming existential crises which are threatening human extinction: or to the inescapable fact that, the kind of problem-solving power maths can muster, will be urgently needed to solve them.

The gist of the former Blog was that the maths hierarchy is still fixated on searching-for elegant, abstruse-ornamental conclusions.  These remain as the implied glorious end-product of the subject.  But fellow mathematicians must be, at least slightly, aware that this looks a little quaint:  they know that most of today’s educated public have lost their previous excessive awe of the subject.  But leading maths gurus are sticking to their guns… in spite of this ominous loss of support.  They know that society has been transformed by the computer revolution… and left in a condition where it (society) is now 100% dependent on maths. So maths is still important. Actually instrumental maths has become extraordinarily important.  And the maths-driven software which runs the modern world needs a cadre of perceptive experts to ensure that it is properly used and understood. Where are they? They will only materialise when maths teaching in schools begins, once again, to captivate the imagination of the brightest students.

Unfortunately the computer sector seems to be doing everything in its power to play down mentions of, references to, or plans for, maths. Today the word ‘mathematics’ is rarely heard. The assumptive stance of the IT companies seems to be to try to create the impression that the potency of their machines derives entirely from modern electronics. Recognising that much of the magic stems from mathematics, would, they feel, detract from the lustre of their gadgets.

A lot of the maths used in today’s software consists of classic results discovered in the 18^thand 19^th centuries. It has to be serviced and modified consistently when new versions of the software are being launched.  This implies that we need a stable workforce of responsible, mathematically-literate experts… to ensure that it is serviced properly, used wisely, and fully understood. The simplistic populist assumption that <<maths is no longer needed>> could not be more wrong. Maths taught in school remains at the heart of 21^st century thinking. We still need a thriving, energised mode of school maths… to replenish the cadre of relevant experts… as old age and retirement takes its toll.

But there is a major $64 question surrounding the basis and philosophy of today’s school maths.  There still has been no unmistakable public signal that maths has outgrown its former Platonic mindset.  The Platonic interpretation of what constitutes the essence of maths was based on the conviction that <<Only the timeless is real>>.  At its best, this was naïve, because everything in the natural world is transient. Also our mortality ensures that no one can ever know if this is true. (A person would have to live for ever to check it.)  And this fixation on timelessness is actually merely based on the simple observation that the truths of maths are time-independent.

This might seem like a way of saying that maths is timeless, but it overlooks the fact that the stuffing of the subject is human reasoning. If there are no mathematicians in the future, there will be —at that time, say the 22^nd century— no recognised mathematic truths. If a few people in 2125 try to find solace in the thought that their predecessors “recognised mathematic truths a century ago” they will have to explain away why no mathematicians survived to carry-on the tradition.

We urgently need to ask ourselves the $64 question:

How “glorious” can we regard abstruse-ornamental maths to be —as a cultural fact— when it is compared with the serious-instrumental maths which now underlies every aspect of human striving?

The answer can only be: very little.

For many, this is a hard saying, but the holy grail of searching for abstruse, ornamental maths —which reigned supreme for more than 2,000 years— can no longer be the main justification for doing maths.

There was, actually, an earlier, hardly noticed, similar exit —when the cult of Temple geometry faded-out in Japan.

Classical musicians might try to shore-up their discipline with the observation that their scores are printed, and hence time-independent. But this kind of time-independence offers only a shadow of satisfaction… compared with the wonderful feeling of lively music.  Similar remarks apply to maths. It is not the dry, written symbolic formulae which encapsulate its essence, but the quality of the live reasoning which it brings.

There can be much wailing about this situation, because it is not funny to discover that a formerly revered quest —abstruse-ornamental maths— has clay feet.  But the great body of classic mathematics remains, and the situation is no worse than it was in 1900 when the gurus of higher maths effectively decided, in their wisdom, that henceforth higher maths was going to be conceived as an “unique intellectual artform suis generis”.

This was disastrous: an own goal, because it turned out that the educated public didn’t value maths for its dry conclusions, still less for the advanced processes it couldn’t understand. The gurus who rooted for aesthetics at the time were unaware that the educated public held maths in high regard mainly for its illuminative power, not for its hyper-abstruse conceits. (The gurus were by then a subgroup which had cut themselves off from the ordinary world. This prevented them from seeing that ordinary people at the time supported maths, and treated it with great respect, because they were aware of the subject’s beneficial effect in the real world… not because it occasionally turned-up Faberge-type gems of (for them) unintelligible significance.)

Maths remains a fascinating, elegant subject based on rigorous logic… but now with the added importance that it is vitally needed to secure the future of our civilisation.

So a metaphorical baton has to be passed-on to schools…to build a generation of youngsters fascinated by mathematics in the new style —i.e. maximising its illuminative power. Students need to be introduced-to, and led to digest, the concepts needed to understand how maths underpins modern progress.  They need to be brought to see that instrumental maths is the central, vital pathfinder for new ideas… something which is already widely valued in innovative circles.  The kind of illumination it delivers is urgently needed wherever new systems are being considered… and the light it can shine onto the predictable implications of different options is priceless.

This is a fundamental change of culture, which needs to be posted, loud and clear.  It is pathetic —-not to say, inexcusable— to be continuing to laud an outmoded, obsolete illusion of what is most worthwhile in maths.

            We urgently need a major statement to this effect from the powers that be.

            In this way maths can become the most fascinating school subject, with a capacity to explore the predictable implications of interesting innovations in situations of many different kinds.

CHRISTOPHER ORMELL November 1^st 2024.  If you would like a free online copy of the P E R Narrative Maths Manifesto, send an email requesting this to per4group@gmail.com  Also comments on the reasoning in this or earlier blogs in this series can be submitted by email to the same address.  This includes any counter-argument submitted as a bid for the prize offered in blog 49.  >> chrisormell@aol.com