Today morale in most local communities is probably at an all-time low. We have been wallowing in gloom since the conjunction of Covid 19, the dreadful 2022 Ukraine War, the poor level of post-Brexit leadership, and the severe austerity caused by outbreaks of strikes and inflation. To a degree, Christmas 2022 shielded us against the sheer rawness of the outlook. But now, as the new year beckons, we are confronted once again with some stark realities.   

So is there a way of peering into the future and seeing good times ahead?Yes, paradoxically, the future is both extremely bright and extremely dark.We won’t get to enjoy the exceedingly bright future, though, unless we can first see-off the various existential threats and crises which are ominously hovering around.

Some of the brightness of the future can be seen in the possible breakthrough pulled-off at UC Berkeley with their new fusion machine, and another breakthrough elsewhere with the new DNA editing method for attacking cancers. But of course the big picture is that two monstrous fudges in the 1920s cast such a spell of deep intellectual gloom onto the human race… And at last these have been, very belatedly, rendered redundant by genuine solutions to the problems which they so inexcusably fudged. The first was the relativity of photons,  which was supposed to be solved by adopting spacetime … about as absolute, static, timeless, and unrelativistic an idea as it would be possible to imagine. The solution is that we need to become aware that mathematics is not the only way to model physical reality. There is also anti-mathematics which is the study of the logic of extended fully transient reality. Anti-mathematic models of physical reality have relativity built-in from the beginning. 

The second was ZF set theory, also as brazen a fudge as it is possible to imagine, because no attempt was made to justify its brusque rejection of transparent set theory, and its arbitrary replacement by ad hoc opaque set theory. The problem was that Russell’s Contradiction (1901) showed that transparent set theory lacked some vital checks and constraints needed to prevent a total meltdown of logic.  These vital constraints, it is now clear, are commonsense rules to avoid dynamic contradiction, which becomes possible when a set can be a member of itself… which it obviously can.  The higher maths gurus of the 1920s tried to place all the blame on the concept of set self-membership. But ‘the set of all sets mentioned in this blog’ is quite obviously a ‘set mentioned in this blog’, so it involved denying the obvious.  So the effect of this silly, unwise ploy was to show the world that the gurus of higher maths in the 1920s had lost all trace of an awareness that mathematics must be built on transparency. 

At the time they got away with it.

Hardly anyone at the time was aware that these questionable assumptions magisterially adopted in the 1920s were brazen, inexcusable fudges, still less that they would eventually have the effect of universally downgrading —almost extinguishing—  logic, maths, clarity and rationality.  

Today this long-standing, deadly blight on reason and rationality can be lifted.  But the number of people who can see these ‘lifting’ truths is limited, because the higher maths leadership imposes its recommendations onto the professional maths community with an iron fist.  This is a very bright light, but it is only going to be seen at first by those who have learnt to take the recommendations of the maths leadership with a pinch of salt.

A lesser, but practically potent, bright light is a new way of teaching mainstream school maths… via rich narrative scenarios… which make palpable sense. They stimulate student curiosity by posing visibly awkward real-life questions which can only be satisfactorily solved using basic maths. This was the theme of a Manifesto issued by the PER Group at a meeting in Conway Hall, London in October 2022.  It proposes a profound reform, which can rebalance school maths in a way which has actually been needed for more than twenty centuries. 

For this very long time the gurus of higher maths have laid down how maths should be taught in schools. They were the crème de la crème of the intellectual elite and their considered opinion was almost universally respected.

Now they have been comprehensively humbled by the ascendancy of IT and their own mistakes, conditions have arisen which allow intelligent lay opinion to recognise that what has been offered in the past as the “maths curriculum” is a badly damaged, broken assemblage of raw symbolic challenges which make no immediate sense at all to the average student.  

They look esoteric and pointless to the average student.

In ordinary language the word ‘eight’ is an adjective which tells the listener  something about the size of a small collection of things. But in maths the word ‘eight’ has been somehow promoted to be a noun, a ‘subject’, a ‘thing’ a focus of attention. Only the gurus of maths see it like this. The average person —and her or his children— naturally see it as adjectival, and quite obviously not as any kind of ‘thing’.

In the past, and for countless centuries, the average person was browbeaten into a recognition that the gurus were right. But this insistence came at a price.  As a result the “average student” tended to feel that she or he was a lifelong “mathematic failure”. 

But now the gurus of higher maths have lost their former stellar authority, and this changes the situation dramatically. It makes maths-as-a-subject seem arbitrary, useless and redundant. 

We are reaping the dangerous results of this downgrading of maths in a widespread loss of rigour and discipline in many practical situations.

However maths began as a simple tool for handling groups of things and layouts. So for many centuries of pre-history it was an instrumental subject and one which was highly valued.

We need to return to this purposefulness… in a way which makes sense for ordinary people. 

Rich narrative scenarios which “bring out” this purposefulness are possible for all topics of the mainstream maths curriculum —this was established by the 10-year Schools Council Project Mathematics Applicable which ran at Reading University from 1969 to 1978. The present author was its Director.  The Reading Project only covered the last two years of schooling, but the same principles apply —and are easier to implement— at the earlier age levels.

It will take time for a large archive of such stimulating scenarios to be devised, but the potential for such scenarios remains secure.  The small minority in each school cohort which can happily handle uninterpreted maths dilemmas can, of course, still be taught separately, using schemas based on oxygen-free, rareified, raw symbolic challenges.  

So these bright lights exist, if only at the present time as potentialities.  But in the foreground we have a great deal of darkness, associated with the existential crises of global warming, truth-blurring, IT-scamming, relationship breakdowns, dysfunctional education, mental health breakdowns and substance abuse.

Are there solutions to the existential crises?  Yes, but they are all predicated on large numbers of sensible, responsible people turning over a new leaf and realising that we are trying to operate today’s civilisation without the minimal internalised ethical standards which, in the past, were always judged to be essential.  Without minimal ethical standards treated as non-negotiable norms, things will fall apart.  This blinding insight arrived about 7,000 years ago with the first urban centres. It is equally valid today.

CHRISTOPHER ORMELL 1^st January 2023