These blogs are offered to show the way in which conceptual, reasoning-led maths can generate interesting, sometimes pivotal, results. In an age of awesomely microscopic, awesomely fast, massive, digital electronics, the manipulation of symbolic configurations will —of course— be mainly handed-over to computers. We can no longer expect to distinguish ourselves by finding sophisticated manipulations. Automated maths on computers will, of course, always be able to out-do the human manipulator.

But where does this maths come from?  It has to begin as putative reasoning, in the heads of people who are acutely curious about formal coincidences, i.e. mathematicians.

The arrival of ultra-reliable computers —using solid state physics— in the early 1960s seems to have provoked an adverse reaction from the maths establishment. They didn’t see these new machines as welcome aids-to-manipulation, but rather as a hostile invasion which put tanks on their elegant abstract lawn.  For more than a century these maths gurus and their predecessors had been building an immense ‘aesthetic empire’ of higher maths.  (It was supposed to be a superb ‘aesthetic intellectual artform’.) The arrival of computers was initially vehemently opposed because the gurus realised (correctly) that it was going to trample all over this vast estate of elegant, unworldly forms.

So the gurus of higher maths responded to the computer revolution with furious opposition. They forgot that the new machine was the brainchild of Alan Turing and John von Neumann, two mathematicians of the highest calibre. They saw (correctly) that these computers would tip the main emphasis of the subject (in intelligent public conversation) back towards utilitarian and scientific ‘applications’.

This was a savage luddite rebellion. For 130 years the pursuit of extra abstract, extra-esoteric, elegant higher maths had been the name of the game. ‘Modern mathematics’ had been portrayed as theintellectual mystique: the mystique which offered the highest, most marvellous, most sophisticated, order.  Sadly its pioneers had become a law-unto-themselves… and triumphant to the nth degree. Now, all this cumulative empire-building was under existential attack. It was in danger of becoming a footnote in history.

So the gurus stood their ground and counter-attacked.

They devised a bold plan to try to turn school maths into the elementary study of sets. This would become the new focus, taking over from the traditional emphasis on number.(They were blissfully unaware that this looked —from an ordinary point of view — like the dumbest reason for doing maths.)  They were doubling-up on their own aesthetic angles, switching to an emphasis on unusual logics and ever higher abstractions.

This counter-revolution crashed. It all turned out to be a disastrous mistake.

Modern maths for schools failed abysmally, and part of the collateral damage was that the public credibility of the subject —which had formerly been sky-high— caved in. The previous aesthetic 20^thcentury approach to higher maths began to look like an ego trip. Disastrously though (for the lay majority) this fierce aversion to maths quickly spread… into a broad rejection of rationality, theory, philosophy and intellectual thinking generally.

This was the Post-Modern Age. Raw feeling trumped thinking, irrationality burgeoned, and the overall catchphrase was <<Anything Goes!>>.

So after the computer revolution, the objective power of maths increased a thousandfold, but, ironically, the public reputation of higher maths nosedived.

The now cock-a-hoop computerists could have thrown higher maths a lifeline. But after the fierce rejection they had felt, this was of course unlikely.

Since then the computer establishment has done everything in its power to downplay the status of maths.  They started calling mathematical modelling ‘computer modelling’, a major category mistake, because a computer —being a gadget— doesn’t “do anything of its own accord”. (It implements the instructions it has been given by a human modeller.) The computerists told their salespersons to go round saying <<Computers have nothing to do with maths!>>, one of the biggest lies in modern history.  They knew that maths was under a cloud, and was dreaded by a largely mathsphobic populace.

Let’s look at the positives. 

Digital machines have been available since the 1960s to take over the “doing” side of maths.  This is not —as often portrayed— a distressing loss… nor indeed is it the whole story.  It frees the mathematician up, so that she or he can think. And it is inspired thinking which makes the difference in maths.

So the net result of the computer revolution has been, paradoxically, to send higher maths hurtling down… ending eventually in the doghouse.  This upshot can’t be right.

This is a situation where rationality has gone AWOL, and a new, more synoptic, more logical, approach is needed. First, we need to understand, in a much more focused way than before, why we do maths, what it means, how it makes sense, and what it can do for the human race.  Second, we need to teach youth how to handle maths, how to recognise when it can make a serious contribution, how to organise the doing (the necessary manipulations), and finally understand what the conclusions mean.

Today’s world has become a dim, foggy world heavily dependent on maths.  (It may have been somewhat overdone, because when maths is treated as the dominant discipline, it can induce paralysis and over-regimentation. Today there are sectors where it would be more sensible, civilised and humane to allow human beings to use their nous instead.)

The moguls of Silicon Valley have done everything they possibly can to make sure that the contributions made by maths to computer applications are invisible to the public. They have thoroughly airbrushed them out.  The moguls want to maintain the myth that it is their marvellous electronics which creates the magic. They fear that if the public knew that the magic of computers arose almost entirely from mathslike reasoning, they might not sell as many gadgets.

To say that electronics is responsible for the magic of computers, is like insisting that a bicycle is responsible for winning the Tour de France… forgetting to mention that there was a person who actually steered it round the corners, pedalled the course, and, who, after strenuous effort, kept composure. Modern electronics is, we know, a great success.  But its success is on a similar level to airtravel, modern medicine, vegan cheese, modern paint, plumbing, electric toothbrushes…

The great mystery is why the gurus of maths —the former guardians of the discipline— let this happen. Why did they, in effect, abdicate responsibility for automated maths when digital computers first came along?

They seemed happily to waive away the entire applicative thunder of maths, something which left school maths teachers naked, without clothes… deserted, haplessly shivering in the wind.

Fortunately this anti-computer attitude in maths which originated in1960 has long since faded away. Subsequent younger generations have accommodated the computer into their concept of maths, the major step being the launch of Mathematica software (1988). But the spectacular power of Mathematica has not been sufficient to challenge the deeply reactionary quasi-Platonic bias of the ancient subject.

The unnecessary underlying misunderstandings of the 1960s are still in place. They have not yet been resolved.  A new era of fascinating modelling-minded maths awaits. It can be much, much, more interesting for young people in schools.

CHRISTOPHER ORMELL around December 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.