The theme of last month’s blog was a review of disappointments: a list of six mistakes made by the leadership of mathematics, plus reflections on their consequences, and a suggestion for remedying the demoralising effect of Ulam’s Dilemma.  These historic ill-judged decisions of the leadership resulted in maths losing big chunks of its former public credibility. (They were, of course, also serious setbacks for the good name of reason in the eyes of the public.) The suggestion was to reign-in the over-diversification of novel concept-minting which led inevitably to Ulam’s Dilemma.  The proliferation of so much “bizarre, unexpected innovation” flowed from the Artform Turn a new, aesthetic interpretation of the raison d’etre of modern higher mathematics, which was wheeled-in by the gurus of 1900.

So this month we look at new perspectives which might gradually re-build the credibility and integrity of the higher subject, and strengthen its core meaning from the public point of view. 

But there is also a deeper question:  what is at the root of these mistakes?  Why has the subject staggered in modern times from mistake to mistake?  Where is the pathway which can lead us towards better standards of rigour and clarity-of-thought in mathematics?

When Reuben Hersh chose the title of his impressive last book What is mathematics really? (1996) he the nail on the head. This is the $64 question. Mathematics first emerged in a form recognisable today as “more than mere arithmetic” during the golden age of Greece. But there was already a large body of arithmetic truths and algorithms in existence, not to mention geometric truths like those used to design and build the pyramids. And all this pre-existing know-how had emerged in an era dominated by religion. So the interpretations of the meaning of mathematical truths adopted in these early stages of the growth of the subject were naturally coloured to a degree by something (religion) which we can see today has nothing to do with mathematics. Mathematics was generally treated in the Ancient World as the very special (‘eternal’) language used by the God who, it was thought, must have created the universe.  

On a more mundane level, mathematics offered a uniquely satisfying kind of intellectual problem-solving which was fully known to, and enjoyed by, Greek geometers.  And the truths they discovered as a result of this cognitive activity were not tautologies, but capable of surprising any attentive observer.  That every triangle has nine particular points which always fall on a special circle is not a truism,  Pythagoras’s theorem is not a truism, that 2 can never be expressed as a ratio of two integers is not a truism.

We know that the brotherhood of Pythagoras considered that <<they had discovered that God made a mistake when he created the universe>>, because everything had, of course, to be proportional to everything else. But the diagonal of a unit square could never be proportional to the side of the square!

This was dynamite.

No one today would draw such a shocking religious conclusion from a similar train of reasoning. 

Those who practised such geometric problem-solving were, of course, only a tiny fraction of the Greek population: they were the ones sufficiently able, and sufficiently well-heeled, to be able to do so. But they were lionised by some ordinary Greek citizens, because their subject, geometry, was widely regarded as a very useful as well as a very civilised, important craft. 

This attractive view of the vocation of the pure mathematician has stayed more or less unchanged ever since.  G. H. Hardy expounded it eloquently in his Mathematician’s Apology written during WW2. But there has been a change since Hardy’s day. When Hardy wrote his book, pure mathematics was still, as he explained, the most useful part of mathematics. It was useful because pure mathematics was needed to solve the equations conceptualised by applied mathematicians working on real-world problems. Without the priceless technical know-how which could only be provided by pure mathematics, key equations would be left unsolved, and much mathematical modelling was almost useless. Only if you could solve the relevant equations, could you know what the model was telling you.

(Hardy, though, conveniently glossed over the fact that in 1940 most of the pure mathematics which had been explored in the previous decades was in regions far removed from the mathematical modelling being done by applied mathematicians.)

It was this extreme “usefulness” of less than a 1/1000 th part of the body of known pure maths, which sustained the credibility of pure mathematics as a whole.  Pure mathematics was like mining diamonds.  Many tons of material had to be processed so that  —every now and then— a tiny bit of priceless technical know-how could be revealed… of the kind needed to solve modelling equations.  

But in 1940 this was about to change. Not many miles away at Bletchley Park, Alan Turing, with help from the specialist engineer, Tommy Flowers, was going to organise the first electronic computer, the Colossus.

When electronic computers became unbelievably reliable (around 1960), they began to offer powerful heuristic ways to solve equations —methods which could be applied to almost any equation or set of equations. The ‘solutions’ found were often only approximations, but this was all that is needed in most technological and scientific situations.

So the tenuous thread, on which pure mathematics had been relying for its credibility, had snapped.

The former pricelessness (in useful terms) of its special knowledge had gone, and henceforward it was going to be difficult to justify financing research in pure mathematics from the public purse. 

This was the key background to the glitch (No 6) which precipitated the worst ever nose-dive in mathematics’ public credibility. The 1960s revolution in school maths was, in effect, a desperate gamble to secure higher mathematics’ significance in the opinion of the educated public, the politicians, etc.. 

Why did it fail?  Because it revealed that the higher maths establishment hadn’t the faintest idea how their subject was accepted, valued and understood by the public or the public’s children. G. H. Hardy’s percipience was had been forgotten. Those who backed the revolution had no idea that their vision of what was valuable in maths was miles away from that of the average person.

A similar howling misjudgment of public attitudes had been made by the gurus of 1900 when they almost unanimously made their Artform Turn.  They were undermining their own public credibility, but they didn’t realise this.

In other words, it is all very well claiming that truths in higher maths can be justified “as ends in themselves”, but this only carries conviction with the educated public when there is a background, useful element as well. Once the useful element has clearly gone, the claim is no more credible than a similar claim for any other kind of pure ornamentation.

This is the modern situation.

But… the religious mystique wrongly associated with maths also gave the “end in itself” argument some extra clout. It worked well during the ages of Belief. But we are clearly now in a Post-Christian era, and that vital extra clout has gone.

So is everything hopeless?

No, not at all.

Looking back to the earliest stages of mathematics, the gurus of the subject claimed that they were the only people who really understood it, and what its conclusions meant. They could do this with conviction, because they had shown that they were the only people who had sufficient ‘iron in the soul’ (a phrase of Rosza Peter) to persist with mathematical inquiries which were abstract mumbo-jumbo to the average person.

But even in the 6^th century BCE, their dismissal of the notion that the meaning of mathematics “came in any way from its uses” was a mistake. Yes there were millions of trivial, mundane uses. But there were also modelling uses, which gave Emperors, Princes, Generals, etc. valuable (essential) previews of their putative military projects. Here a big bundle of arithmetic gave a three-dimensional picture of the predictable features of a proposed battle with very important synoptic implications. (Were there enough swords? Was there enough fodder to feed the horses? etc.) Getting the logistics right, could dramatically swing the result of a conflict. This was the real source of the social status accorded to the leaders of mathematics for the following 25 centuries. The Emperors, Princes, Generals, etc. who valued modelling mathematics were, let’s face it, mostly thugs. They didn’t accord their mathematic advisers great respect because they appreciated the mathematic skill involved, but because the overall result of their work could be a surefire way to win a war.

We have been living through the most remarkable half-century of progress in modelling mathematics. Amazing feats have been achieved, such as the analysis of DNA, and bringing back astronauts from the Moon using only a few litres of fuel.

It is unfortunate, therefore, that the leadership of mathematics let the computer industry claim all the credit in the eyes of the public for these brilliant successes —because the leadership made no effort to contest the self-serving claim of the computer fraternity that they were “down to computer modelling”. (Another glitch not listed in Blog Part 6.)

Of course any sensible person today turns to the computer to implement his or her mathematical modelling. This no more means that “it was the computer which did the work”, than it would be to say that the Tour de France was “won by the bicycle which did the work”.

Recently a startling development has been unveiled   —Actimatics which is a completely new mathematical science based on jumping-random tallies instead of static tallies. (For details, see the companion blog philosophyforrenewingreason.com) The good news is that progress in Actimatics will depend largely on finding solutions to difficult quasi-pure mathematics. This could be described as an ‘area of applications’, but these are applications related to symbolic combinations, so in practise they look much more like typical higher maths. 

So the answer to the question <<What was the root of the trouble?>> was that for more than 2,500 years the maths fraternity claimed to be the only group of people who understood the meaning of mathematics.  But they didn’t.  Their vision had been undermined by a constant need to cut out real-world appraisals, to peer at abstruse abstractions. And this had had the unnoticed effect that it precisely blocked the kind of mental skills needed to see mathematics in the round, as it really was, out there… a human institution which gives society priceless insights into the future.

CHRISTOPHER ORMELL November 2020